Question: Simplify; express your answer in exponential form. Assume $q\neq 0, y\neq 0$. $\dfrac{{(q^{5}y^{4})^{2}}}{{(q^{-5}y^{-2})^{4}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{5}y^{4})^{2} = (q^{5})^{2}(y^{4})^{2}}$ On the left, we have ${q^{5}}$ to the exponent ${2}$ . Now ${5 \times 2 = 10}$ , so ${(q^{5})^{2} = q^{10}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{5}y^{4})^{2}}}{{(q^{-5}y^{-2})^{4}}} = \dfrac{{q^{10}y^{8}}}{{q^{-20}y^{-8}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{10}y^{8}}}{{q^{-20}y^{-8}}} = \dfrac{{q^{10}}}{{q^{-20}}} \cdot \dfrac{{y^{8}}}{{y^{-8}}} = q^{{10} - {(-20)}} \cdot y^{{8} - {(-8)}} = q^{30}y^{16}$